I thought David Ross's comment on BT and Borel sets was pretty clear;
let me attempt a paraphrase. The existence of paradoxical decompositions
has two sources: the geometrical complexity of R^3, as witnessed by the
fact that F_2 embeds in its isometry group, and the richness of P(R), as
witnessed by the existence of a wellorder of R. If you restrict your view
to Borel sets, you can see neither a wellorder nor a paradoxical
decomposition, and if the sight of such offends you, then I suppose that's
some advantage to you. Ross himself found the sight inoffensive. In any
case, you have made no progress whatsoever in taming the geometrical
complexity of R^3 coming out of the fact that F_2 embeds in its isometry
group.
Is this an adequate paraphrase?
There is a larger question here: how is TBU meant to be used? One
can restrict one's attention to Borel sets in the contexts where it is
appropriate to do so without having a formal theory like TBU around.
People have been doing it for the last 100 years or so.
John Steel